Finite Sample Analysis Of Dynamic Regression Parameter Learning

We consider the dynamic linear regression problem, where the predictor vector may vary with time. This problem can be modeled as a linear dynamical system, with non-constant observation operator, where the parameters that need to be learned are the variance of both the process noise and the observation noise. While variance estimation for dynamic regression is a natural problem, with a variety of applications, existing approaches to this problem either lack guarantees altogether, or only have asymptotic guarantees without explicit rates. In particular, existing literature does not provide any clues to the following fundamental question: In terms of data characteristics, what does the convergence rate depend on? In this paper we study the global system operator -- the operator that maps the noise vectors to the output.